On 10/28/06 at 5:21 AM, banerjee.28 at osu.edu (Bonny Banerjee) wrote:
>Is it possible for Mathematica to solve this problem:
>Given sets A and B, does there exist a function from A to B? If yes,
>what is the function?
>Here is an example:
>Let, A = {x such that 0<x<11 and Mod[x,2]==0}
>B = {y such that 0<y<11 and Mod[y+1,2]==0}
>Then, there exists a function from A to B
>y = x - 1
>Thus, is there a way to specify arbitrary sets A, B, and use
>Mathematica to figure out whether there exits a function from A to B
>or not?
Yes and no. Yes, in the sense that for any finite data set there
is always a function that exactly maps members of A to B. If
nothing else, a polynomial of degree Length[data]-1 will do the
trick though not usually the best choice. For this purpose,
cubic interpolation is usually a better choice
Cubic interpolation can be accomplished using the built in
function Interpolation
But the general problem you are expressing, that is find an
arbitrary function from a finite list of points at which the
function has been sampled has no solution or perhaps more
correctly no unique solution.
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